   Chapter 10, Problem 54RE

Chapter
Section
Textbook Problem

# Show that if m is any real number, then there are exactly two lines of slope m that are tangent to the ellipse x2/a2 + y2/b2 = 1 and their equations are y = m x ± a 2 m 2 + b 2

To determine

To find: Prove that the two lines of slope are tangent to ellipse x2a2+y2b2=1 and their equation is y=mx±(b2+a2m2) .

Explanation

Given:

Equation of the ellipse is given as below.

x2a2+y2b2=1 (1)

Calculation:

Equation of the tangent with slope m is of the form as below.

y=mx+c (2)

Substitute (2) in (1).

x2a2+(mx+c)2b2=1x2a2+(m2x2+2mcx+c2)b2=1x2a2+m2x2b2+2mcxb2+c2b2=1

(1a2+m2b2)x2+(2mcb2)x+(c2b21)=0 (3)

Equation (3) is of the form ax2+bx+c=0 .

a=(1a2+m2b2)

b=(2mcb2)

c=(c2b21)

Discriminant of this equation is b24ac=0 .

(2mcb2)24(1a2+m2b2)(c2b21)=04(mcb2)24(1a2+m2b2)(c2b21)=04[

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